1    THE METHOD OF SUCCESSIVE APPROXIMATIONS       267
PROBLEMS
1. Let A be a compact metric space, d the distance on A, v a mapping of A into itself
such that for every pair (x, y) of distinct points of A, d(v(x), v(y)) < d(x9 y). Show that
there exists a point z e A such that v(z) = z. (Use contradiction, by considering the
number c = inf d(x9 v(x)) and proving that there exists a point y e A such that
Let B be the ball \\x\\ ^ I in the space (cq) of Banach (Section 5.3, Problem 5). Let
u be the continuous linear mapping of (cq) into itself such that
(n ^ 0),
and let v(x) = J(l 4- \\x\\)e0 + u(x). Show that v is a continuous mapping of B into
itself such that for any pair (x, y) of distinct points of B, \\v(x) — v(y) \\ < \\x — y\\9 but
that there is no point z e B such that v(z) = z (use the inequality
n(l-a,)^ l~-i>ifor0^af ^ 1).
i=l	1=1
Let E, F, be two normed vector spaces, u a linear homeomorphism of E, onto a sub-
space w(E) of F; let v : w(E) -» E be the inverse mapping of «, and let m = \\v\\.
 (a)	Let A  be  an  open  subset  of E, w a mapping of A into F such that
||h>(*i) — w(x2)|| ^ k\\x\ — x2\\ for xi,x2 in A. Show that if the constant k is such
that km < 1, then jc->/(jc) == u(x) -f- w(x) is a homeomorphism of A onto /(A). If
in addition, E and F are Banach spaces, and w(E) = F, show that /(A) is an open
subset of F (use (10.1.3).)
 (b)	Suppose w is a continuous linear mapping of E into F such that ||w|| < l/m.
Show that / is then a linear homeomorphism of E onto /(E). Furthermore,
for any y0eu(E) such that lb>0|| = l, show that there exists yef(E) such that
lly — Jo 11 ^ wi||w||; conversely, for any ysf(E) such that ||.y||= 1, show that there
exists y0 e u(E) such that \\y — yQ\\ ^ wi||w||/(l — /w||w||).
Let E, F be normed spaces, u a continuous linear mapping of E into F, such that
N = w~1(0) is a finite dimensional subspace, then there exists a closed topological
supplement M of N (Section 5.4) such that the restriction of m to M is a homeomor-
phism onto w(M) = w(E) (see [6]). Let w be a continuous linear mapping of E into F;
show that if ||w|| is small enough, and/= u -\- w, then/"HO) has finite dimension at
most equal to the dimension of ""HO), and there is a closed topological supplement P
of/~HO), such that the restriction of / to P is a homeomorphism onto/(P)=/(E)
(use Problem 3(b)).
Let E be a Banach space, F a normed space, u a linear homeomorphism of E onto u(E)
such that there is a topological supplement Q of w(E) in F. Show that if w is a conti-
nuous linear mapping of E into F with sufficiently small norm ||w||, and /= u + w,
then Q is still a topological supplement of/(E) in F. (Show that the projection of F
onto w(E), restricted to/(E), is a linear homeomorphism onto w(E) when ||w|| is small
enough, using Problem 3.)
Let E, F be two Banach spaces, u a continuous linear mapping of E into F such that
N = tt~HO) has finite dimension p; then N has a topological supplement M such that
the restriction of u to M is a homeomorphism into w(M) = w(E). Suppose in addition
w(E) has finite codimension q in F. Let w a continuous linear mapping of E into F;